Impregnable inventory stewardship for a closed loop supply chain besides energy usage, defective production and green investment manoeuvring pentagonal fuzzy number

To enhance the inventory level and investments under a carbon tax policy, a mathematical inventory model for a manufacturer - multi retailer system is recommended. Manoeuvring production rate adjustment the energy engrossed by the production and rework and emissions due to supply chain operations are scrutinized and administered. To toe the line with the carbon policy and to reduce the emissions, the manufacturer subsidizes in green production technologies. This model focusses to curtail the joint total cost by ascertaining the shipment frequencies, shipment lot, safety factor, production rate, collection rate and investment concurrently. Reduction in total cost can be achieved through emission control, energy consumption and the number of defects by adjusting the production rate. The green investment reduces emissions and increasing returns on used products resulting in an effective environmental performance of the supply chain.